APPLICATION OF LINEAR ALGEBRA IN ELECTRICAL

CIRCUIT

Welcome

PRESENTED TO

Md. Mosfiqur RahmanSenior Lecture in Mathematics Department of GEDDaffodil International University

Presented by

Gazi Md Badruzzaman JHONElectronic & Telecommunication Engineering ID:171-19-1937Daffodil International University

CONTENTS……!

☼Introduction

☼Linear Algebra & various fields

☼History of Linear Algebra

☼Electrical Circuits

☼Electrical circuit In Linear Algebra

☼Gaussian Elimination

☼ The Wheatstone Bridge

INTRODUCTION

• This presentation is mainly about to let us all know that how

electrical circuits works on applications of LINEAR ALGEBRA

• All you need to be a inventor is a good imagination and a pile of

junk.

by:-

THOMAS EDISON

WHAT IS LINEAR ALGEBRA?

• Linear Algebra is the branch of mathematics concerning

vector spaces and linear mappings between such spaces. It

includes the study of lines, planes, and subspaces, but is also

concerned with properties common to all vector spaces.

• Hence, the above definition confirms that Linear Algebra is an

integral part of mathematics.

Abstract Thinking

Chemistry

Coding Theory

Cryptography

Economics

Elimination Theory

Games

Genetics

Geometry

Graph Theory

Heat Distribution

Image Compression

Linear

Programming

Markov Chains

Networking

Sociology

The Fibonacci

Numbers

Eigenfaces

Applications of Linear Algebra in various fields.

LINEAR ALGEBRAHistory:

The study of linear algebra first emerged from the study of

determinants, Determinants were used by Leibniz in 1693

Gabriel Cramer devised Cramer's Rule for solving linear

systems in 1750.

Gauss further developed the theory of solving linear systems

by using Gaussian elimination

In 1844 Hermann Grassmann publish "Theory of Extension“

which founded on linear algebra In 1848, James Joseph

Sylvester introduced the term matrix

Linear algebra first appeared in American graduate textbooks

in the 1940s and in undergraduate textbooks in the 1950s

ELECTRICAL CIRCUITS

† Electrical circuit is nothing but just a combination of

transistor, capacitor, diodes, etc. including some logic gates.

† Each component has it’s own specification.

† And through which we get to know what currents and

voltages are.

† An electrical circuit is a path in which electrons from a voltage

or current source flow.

LINEAR ALGEBRA IN ELECTRICAL CIRCUITS

• Linear Algebra most apparently uses by electrical engineers.

• When ever there is system of linear equation arises the

concept of linear algebra.

• Various electrical circuits solution like Kirchhoff's law , Ohm’s

law are conceptually arise linear algebra.

GO ON…

• To solve various linear equations we need to introduce the

concept of linear algebra.

• Using Gaussian Elimination not only computer engineers but

most of daily computational work minimized .

• Now we don’t have to use extremely large number of pages to

calculate complex system of linear equations.

GAUSSIAN ELIMINATION

To fix all the assertion that we have performed earlier we use

Gaussian elimination.

In this method we need to keep all eqs. into matrix form, for e.g.

Since the columns are of same variable it’s easy to do row operation

to solve for the unknowns.

GO ON…

This method is known as Gaussian Elimination. Now, for large

circuits, this will still be a long process to row reduce to echelon

form.

With the help of a computer and the right software , the large

circuits consisting of hundreds of thousands of components can be

analyzed in a relatively short span of time.

Today’s computers can perform billions of operations within a

second, and with the developments in parallel processing, analyses

of larger and larger electrical systems in a short time frame are

very feasible

THE WHEATSTONE BRIDGE

• The next application is a simple circuit for the precise

measurement of resistors known as the

Wheatstone Bridge. The circuit, invented by Samuel Hunter

Christie (1784-1865) in 1833, was named after Sir Charles

Wheatstone (1802-1875) who ‘found’ and popularized the

arrangement in 1843. It consists of an electrical source and a

galvanometer that connects two parallel branches, containing

four resistors, three of which are known. One parallel branch

consists of a known and unknown resistor (R4), while the other

branch contains two known resistors.

• Kirchoff ’s Current Law yields:

• I0 - I1 - I2 = 0

• I1 - I5 - I3 = 0

• I2 + I5 - I4 = 0

• I3 + I4 - I0 = 0

• And Kirchoff ’s Voltage Law yields:

• I2R2 - I5R5 - I1R1 = 0

• I5R5 + I4R4 - I3R3= 0

• I2R2 + I4R4 - E = 0

• I1R1 + I3R3 - E = 0

In this case, we observe a

circuit that has a 5-volt

power supply with

different loops, and its

resistors.

Notice now that we have three loops

drawn, all rotating clockwise. Next,

we must drawn loops in which the

current in the circuit travels, called

I1, I2, and I3. I1, I2, and I3 are all

current loops (measured in Amps).

n

∑ In *Rn=Vn=1

We start with the general equation

Where V is the voltage, I is the current around a loop, and Rn is the total

resistance of the path for the given current In.

Next, we want to look at each loop, and set up an equation, which uses

all paths that touch the loop multiplied by their total resistances where

they touch that path. Observe the following equations:

18I1 – 2I2 -5I3 = 5

-2I1 + 5I2 -3I3 = 0

-3I1 – 5I2 +9I3 = 0

The coefficients for I1, I2, and I3 are all the

total resistances for those loops, which have

unknown current, and they are set equal to

the total potential difference (voltage) around

that loop. We can then put these equations

into an augmented matrix and put the matrix

into rref

18 -2 -5 5

-2 5 -3 0

-5 -3 9 0

1 0 0 0.4215

0 1 0 0.3864

0 0 1 0.3630

When we put the system is put into an augmented matrix, we get the following:

When we row reduce this matrix, we get

From this, we can

determine what the

current through I1, I2, and

I3 are.